Term Rewriting System R:
[x, y]
minus(minus(x)) -> x
minux(+(x, y)) -> +(minus(y), minus(x))
+(minus(x), +(x, y)) -> y
+(+(x, y), minus(y)) -> x

Termination of R to be shown. 



   R
     ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

minus(minus(x)) -> x
+(minus(x), +(x, y)) -> y
+(+(x, y), minus(y)) -> x

where the Polynomial interpretation:
  POL(minus(x1))=  1 + x1  
  POL(minux(x1))=  2 + x1  
  POL(+(x1, x2))=  x1 + x2  
was used.

Not all Rules of R can be deleted, so we still have to regard a part of R. 


   R
     ↳RRRPolo
       →TRS2
         ↳Removing Redundant Rules



Removing the following rules from R which fullfill a polynomial ordering:

minux(+(x, y)) -> +(minus(y), minus(x))

where the Polynomial interpretation:
  POL(minus(x1))=  x1  
  POL(minux(x1))=  1 + x1  
  POL(+(x1, x2))=  x1 + x2  
was used.

All Rules of R can be deleted. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳Overlay and local confluence Check



The TRS is overlay and locally confluent (all critical pairs are trivially joinable).Hence, we can switch to innermost. 


   R
     ↳RRRPolo
       →TRS2
         ↳RRRPolo
           →TRS3
             ↳OC
             ...
               →TRS4
                 ↳Dependency Pair Analysis



R contains no Dependency Pairs and therefore no SCCs.

Termination of R successfully shown.
Duration:
0:00 minutes